Jeremy Côté

Bits, ink, particles, and words.

Cramer's Rule for Solving Linear Systems

Throughout secondary school, we learn about solving simple systems of equations. We learn that there are several methods (which are more or less the same thing): comparison, substitution, and elimination are the big three. We then go through tons of practice questions which all focus on doing this kind of solving. In particular, we solve systems of two equations, and I don’t recall ever doing more than that. The sense I get from students is that solving these kinds of equations is sometimes confusing. I personally think it’s because there’s a lack of equivalence between the methods, so they all seem like pulling magic tricks out of a hat. That’s a pedagogical/time problem, but I don’t want to focus on that today. Instead, I want to focus on something that makes solving these systems of two equations super quick. It’s called Cramer’s rule, and this method makes it possible to forego using comparison, substitution, or elimination to solve for variables. Instead, you apply this procedure, and you can simply read off the answer.

The Distance Formula (Equation Explainer)

The distance formula is one of the most frequently used relations in physics, allowing us to decompose a variety of vectors into different components. It’s something that every physics student uses, and so it becomes second-nature for most of us. However, I’ve come across the sad fact that many secondary schools don’t seem to teach how the distance formula comes about and its connections with earlier work. As such, the link between algebra and geometry is lost and the distance formula gets lost in calculating differences between \$x\$ and \$y\$. Here, the goal is for us to look at the distance formula and see how it relates to other concepts that are of much use.